Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #2404 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9,14,17)(8,11)(10,12)(13,15,19,20)(16,18);(5,7,9,12,10,14,17,6)(8,13,15,18,16,19,20,11);(2,4)(5,8,10,16)(6,11,12,18)(7,13,14,19)(9,15,17,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #133374, #133375, #133378, … |
| maximal normal subgroup of | #133374, #133375, #133378, … |
| maximal quotient of | #137679, #137694, #137727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6071);